The Tits construction for short $\mathfrak{sl}_2$-super-structures

In this paper, we generalize the Tits construction for Lie superalgebras such that $\mathfrak{sl}_2$ acts by even derivations and decompose, as $\mathfrak{sl}_2$-module, into a direct sum of copies of the adjoint, the natural and the trivial representations. This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the $\mathfrak{sl}_2$-Lie superstructure in terms of $\mathcal{J}$-ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short $\mathfrak{sl}_2$-Lie superalgebra from a $\mathcal{J}$-ternary superalgebra $(\mathcal{J},\mathcal{M})$. Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}.